# Full text of "Treatise On Analysis Vol-Ii"

## See other formats

```90       XII   TOPOLOGY AND TOPOLOGICAL ALGEBRA

£ %n = SUP £« (5.5.1), but the inverse bijection of F onto E is not continuous:

n                n

otherwise F would be complete, and therefore closed in ^R(N) (3.14.4), which
is absurd, because its closure in ^R(N) is the set of all sequences (<!;„) such that
lim = 0.

(12.1 6.1 0) Let E, F be two Banach spaces, and u a continuous surjective linear
mapping ofE to F. Then there exists a number m>0 such that for each * e E
there exists x' e Efor which u(x) = u(xf) and \\u(x)\\ ^m- \\x'\\.

Bearing in mind (12.14.8) and (5.5.1), this expresses that the bijection
F -» E/w"1^), the inverse of the bijection E/w"1^}) ~» F induced by w, is con-
tinuous on F.

(12.16.11)       (Closed graph theorem)   Let E, F be two Frechet spaces. For
a linear mapping u ofE into F to be continuous, it is necessary and sufficient that
the graph (1.4) of u should be closed in the product space E x F.

In general, if /is a continuous mapping of a topological space X into a
HausdorfTtopological space Y, the graph of/is closed in X x Y, because it is
the set of all z e X x Y satisfying the relation pr2z — /(pqz), and the assertion
follows from (12.3.5). To show that the condition stated in (12.16.11) is
sufficient, remark that it implies that the graph G of w, being a closed vector
subspace of the Frechet space E x F (3,20.1 6(iv)), is a Frechet space (3.14.5).
The projection z\-^pr1z of G onto E is therefore a continuous bijective linear
mapping, hence an isomorphism (12.16.9). Since the inverse mapping is
v : #»->(jt, u(x)\ it follows that x\-+u(x) = pr2(v(xj) is continuous on E.

The condition of (12.16.11) may also be expressed by saying that if a
sequence (xn, u(xnj) in E x F tends to a point (x, y), then y = u(x). Replacing
xn by xn — x, and using the linearity of u, an equivalent formulation is that if a
sequence (xn) in E tends to 0 and is such that the sequence (u(x^)) tends to a
limit y, then y = 0. It is this criterion that we shall apply in practice to verify
the continuity of u.

Finally, the following consequence of Baire's theorem allows us to use the
criterion (12.11.5):

(12.16.12)   Let G be a separable, metrizable, locally compact group, acting
continuously and transitively on a Hausdorff topological space E in which every
point has a neighborhood homeomorphic to a complete metric space. For each
xeE, let Sx be the stabilizer ofx. Then the canonical bijection fx : G/SX -+Eisa
homeomorphism.d in the closure of w(W) + w(W) = w(W + W)c w(V),
```